The present invention relates generally to the analysis of the steady-state behavior of circuits driven by a periodic input signal.
The rapid growth in the field of wireless RF telecommunications has provided significant motivation for the development of simulation tools that can quickly and accurately analyze both the steady-state behavior and the response to multiple-frequency excitations of nonlinear circuits. FIG. 1A depicts a generic circuit 10 that is frequently analyzed in engineering applications. Circuit 10 is driven by a periodic input signal 11, u(t), of arbitrary shape, and has an output 12, v(t), that is periodic with the same frequency as u(t) when circuit 10 has reached a steady state. There is a great need for computationally efficient techniques for determining the steady-state behavior of a circuit such as circuit 10, particularly given the fact that frequently the circuit behavior must be characterized over a wide range of input signal frequencies and amplitudes.
One traditional technique for determining a periodic steady-state solution (PSS) is transient analysis. This technique involves determining respective states of the circuit at a series of timepoints (starting at time zero with a set of initial conditions), the state determined for the circuit at any particular timepoint depending on the state determined for the circuit at the previous timepoint. The technique terminates upon detecting that a steady-state has been reached. Transient analysis requires several timepoints per period of the input signal, and thus is often impractical in circuits having a time constant many orders of magnitude greater than the period of the input signal. For example, FIG. 1B depicts a self-biased amplifier circuit 20 that is driven by a periodic input signal 21, u(t). The time constant of circuit 20 might be chosen to be several orders of magnitude larger than the largest possible period for u(t), thereby requiring transient analysis over thousands of timepoints.
Another prior art technique for obtaining the PSS solution for a circuit involves the use of the shooting-Newton method. This method uses a Newton iterative technique to compute a series of estimates for the PSS. The difference between each estimate in the series and the previous estimate is determined by solving a respective linear system of equations via Gaussian elimination. Obtaining the solution for each linear system by Gaussian elimination involves factorization of a dense N by N matrix, where N represents the number of nodes in the circuit, and thus requires a computation time that increases with the third power of N. For this reason, prior art shooting-Newton techniques for obtaining a PSS solution are often impractical for circuits with a large number of nodes.
Another prior art technique for obtaining a PSS solution involves the use of a finite-difference method. Such a method discretizes the circuit differential equations at a set of timepoints spanning an interval of the input signal to obtain a set of difference equations. Solution of this set of difference equations provides a PSS solution for each of the timepoints simultaneously. Like the previous PSS methods discussed, PSS techniques based on finite-difference methods typically have a computational cost that grows as the cube of the number of circuit nodes, and thus also may be impractical for large circuits.
FIG. 1C depicts another type of generic circuit 30 that is frequently analyzed in engineering applications. Circuit 30 is driven by two periodic input signals 31-32, ul and us. us is a small sinusoidal signal of much smaller amplitude than ul. ul is periodic with arbitrary shape. One approach to the solution of circuit 30 is transient analysis. Transient analysis may be a very expensive option, particularly where one of the input signals is of much higher frequency than the other. For example, FIG. 1D depicts a switched-capacitor filter circuit 40 driven by a periodic signal us to be filtered and a high frequency clock signal ul that controls the state of a transistor 41. To ensure accurate results, transient analysis must cover several timepoints per period of the high frequency clock ul, even though the frequency of the filtered signal us is much lower.
Another approach to studying the behavior of circuit 30, known as Periodic Time-Variant Analysis (PTV), treats the small input signal, us, as a small perturbation to the periodic steady-state (PSS) response of circuit 30 to ul when us=0. In particular, typically, a small-signal steady-state response is determined at a set of time points spanning a period of ul, by linearizing the differential equations for circuit 30 about the PSS response, and then discretizing the resulting system at the set of time points. Such a technique has a computational cost that grows with the third power of the number of circuit nodes, and for this reason, is impractical for large circuits.
FIG. 1E depicts another generic circuit 50 of a type frequently analyzed in engineering applications (e.g., mixer circuits). Circuit 50 is driven by d periodic signals 51 of unrelated frequencies, and typically has a steady-state response 52 that is quasiperiodic in d frequencies. One approach to determining a quasiperiodic response involves the use of mixed frequency-time methods as described in detail in chapter 7 of the text xe2x80x9cSteady-State Methods for Simulating Analog and Microwave Circuitsxe2x80x9d by Kundert et al. However, such methods typically have a computational cost that grows with the third power of the number of circuit nodes, and for this reason, are impractical for large circuits.
As discussed above in the background section, various prior art techniques for steady-state and small signal analysis have a computational cost that grows with the third power of the number of circuit nodes, and are thus impractical for large circuits. By contrast, the present invention provides efficient techniques for steady-state and small signal analysis having a computational cost that is approximately linear in the number of circuit nodes. These techniques are, thus, practical for large circuits that cannot be feasibly handled by prior art techniques.
In particular, the present invention provides an efficient method for determining the periodic steady state response of a circuit driven by a periodic signal, the method including the steps of 1) using a shooting method to form a non-linear system of equations for initial conditions of the circuit that directly result in the periodic steady state response; 2) solving the non-linear system via a Newton iterative method, where each iteration of the Newton method involves solution of a respective linear system of equations; and 3) for each iteration of the Newton method, solving the respective linear system of equations associated with the iteration of the Newton method via a matrix-implicit iterative technique.
In one embodiment, the iterative technique is a matrix-implicit application of a Krylov subspace technique, resulting in a computational cost that grows approximately in a linear fashion with the number of nodes in the circuit, in contrast to typical shooting methods used to obtain a periodic steady state response whose computational cost grows with the cube of the number of circuit nodes. Thus, the disclosed method renders feasible periodic steady state analysis of circuits much larger than those that could be analyzed via previous methods. In addition, the invention provides an efficient finite-difference technique for obtaining a PSS solution, the technique having a computational cost that also grows approximately in a linear fashion with the number of nodes in the circuit. An efficient technique that is almost linear in the number of circuit nodes is also provided for determining a quasiperiodic steady-state response of a circuit driven by two or more signals of unrelated frequencies.
The present invention also provides an efficient method for determining the small-signal steady state response to a small periodic signal of a circuit driven by a large periodic signal, the small-signal steady state response covering a plurality of timepoints that span a period of the large periodic signal. The method includes the steps of 1) forming a linear system of equations for the small-signal steady state response at the last of the timepoints; and 2) solving the linear system of equations via a matrix-implicit iterative technique.
In one embodiment, the iterative technique is a matrix-implicit application of a Krylov subspace technique, resulting in a computational cost that grows approximately in a linear fashion with the number of nodes in the circuit. This is in contrast to typical methods which use Gaussian elimination to solve the above linear system and whose computational cost grows with the cube of the number of circuit nodes. Thus, the disclosed method renders feasible small signal analysis of circuits much larger than those that could be analyzed via previous methods. The efficiency of the disclosed method is particularly important given that, typically, a small signal response is required for a range of frequencies.
In addition, the invention provides a technique for reusing (xe2x80x9crecyclingxe2x80x9d), during the determination of the small-signal steady-state response for a particular small signal frequency, matrix-vector products previously obtained during the determination of the small-signal steady-state response for another small signal frequency.